Cameron–Liebler line classes with parameter x=q2−12

Abstract

In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter x = q 2 −1 2 for q ≡ 5 or 9 (mod 12), which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG(2, q), which is closely related to our Cameron-Liebler line classes.

Cameron‐Liebler line classes in PG(3, 5)

“…In this section, we discuss solving the system of (Diophantine) Equations (2) to (10) with respect to the unknowns m w for all ∈ w M, n u for all ∈ u N , and z i for all ∈ i I. Note that this system is not linear with respect to m w and n u .…”

“…We consider the following approach to solving Equations (2) to (10). First, we regard the system of Equations (5) to (10) as a system of linear equations with respect to the unknowns z i (so that its left-hand side consists of linear combinations of z i only), and apply the Gaussian elimination procedure to it, in which n u , m w are treated as indeterminates.…”

“…It may happen that the row-reduced echelon form of the system contains zero equations in its left-hand side with a nonzero right-hand side, which is represented by polynomials in n u , m w of degree at most 2. Therefore, in order the system of Equations (5) to (10) to be consistent the polynomials in the right-hand side of the zero equations must be equated to 0. This may provide additional constraints on a solution of Equations (2) to (4).…”

“…The conjecture turned out to be wrong in q PG (3, ): the first counter example was constructed by Drudge [9], and many more nontrivial line classes have been found during the last decade [2,[5][6][7]10,13,14,16,25]. However, its validity in n q PG( , ) with n > 3 remains an open question.…”

“…• with x = 10 constructed in [14]; • with x = 12 constructed in [25] (later generalised in [7,10], to an infinite family with x = q − 1 2 2 for ≡ q 5 or 9 (mod 12));…”